summation

the sum is sinθ1 + sin2θ2 + ...

Let S = sinθ1 + sin2θ2 + ...
and C = cosθ1 + cos2θ2 + ...

C+ iS = e^iθ + e^2iθ/2 + ...
= - log(1-e^iθ)
= - log(1-cosθ- isinθ)
= loglzl + iarg(z)

So S = arg(z) = tan^-1(-sinθ1-cosθ)

cool soln

adding to the soln

tan^-1(-2sin(θ/2)cos(θ/2)2sin²(θ/2)
= tan^-1(-cotθ/2)
= -tan^-1(tan(pi/2-θ/2)
= θ/2 - pi/2